Hi guys I've been doing some problems on series and I ran into this one,

which I have no idea how to go about doing.

Any help would be appreciated.

Thanks.

Think about what the integral looks like when is large: then you integrate on a very small interval the function , which is almost equal to since is small... and you can integrate explicitly. It gives , which is the general term of a convergent series.

So when n is large the interval [0, becomes small. And this implies that the area bounded by the curve becomes smaller and smaller, and so the series is likely to converge? Hence the comparison with a convergent series?

So when n is large the interval [0, becomes small. And this implies that the area bounded by the curve becomes smaller and smaller, and so the series is likely to converge? Hence the comparison with a convergent series?

Not only is the integral smaller and smaller, but it is like . The idea is to neglect the denominator since it becomes close to 1, so that we obtain a function that we know how to integrate. Actually I almost wrote the full proof. Just use the majoration I gave and integrate it.